Key Concepts
Writing Solutions to Absolute Value Inequalities
For any positive value of a and x, a single variable, or any algebraic expression:
| Absolute Value Inequality | Equivalent Inequality | Interval Notation |
| [latex]\left|{ x }\right|\le{ a}[/latex] | [latex]{ -a}\le{x}\le{ a}[/latex] | [latex]\left[-a, a\right][/latex] |
| [latex]\left| x \right|\lt{a}[/latex] | [latex]{ -a}\lt{x}\lt{ a}[/latex] | [latex]\left(-a, a\right)[/latex] |
| [latex]\left| x \right|\ge{ a}[/latex] | [latex]{x}\le\text{−a}[/latex] or [latex]{x}\ge{ a}[/latex] | [latex]\left(-\infty,-a\right]\cup\left[a,\infty\right)[/latex] |
| [latex]\left| x \right|\gt\text{a}[/latex] | [latex]\displaystyle{x}\lt\text{−a}[/latex] or [latex]{x}\gt{ a}[/latex] | [latex]\left(-\infty,-a\right)\cup\left(a,\infty\right)[/latex] |
Glossary
Union The solution of a compound inequality that consists of two inequalities joined with the word or is the union of the solutions of each inequality.
Intersection The solution of a compound inequality that consists of two inequalities joined with the word and is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an and compound inequality are all the solutions that the two inequalities have in common
